Modeling and Measuring Ferry Boat Wake Propagation

Home , Wake

1 MODELING AND MEASURING FERRY BOAT WAKE

2 PROPAGATION

1 3 Brad Perfect Dr. Jim Thomson 2 Dr. James Riley 3

4 ABSTRACT

5 Surface waves generated by vessels were studied for two classes of ferries in the

6 Washington State Department of Transportation (WSDOT) system. The vessel

7 wakes were measured using a suite of free-drifting GPS buoys. The wakes were

8 modeled using an analytical potential ﬂow model, which has a strong dependence

9 on vessel Froude number and vessel aspect ratio. Both the measurements and

10 the model are phase-resolving, and thus direct comparisons are made, in addition

11 to comparing bulk parameters. The model has high ﬁdelity for the parameters

12 describing the operating conditions for WSDOT vessels, successfully reproducing

13 the measured diﬀerence between the two vessel classes and the dependence on

14 vessel speed for each vessel class.

15 Keywords: Free Surface, Ship Wake, Kelvin Wake.

1PhD Student, Dept. of Mechanical Engineering. Univ. of Washington. 3900 Stevens Way NE, Seattle, WA 98195. E-mail: [email protected]. 2Principal Oceanographer, Applied Physics Laboratory. Univ. of Washington. 1013 NE 40th St, Box 355640, Seattle, WA 98105. E-mail: [email protected] 3PACCAR Prof. of Engr., Dept. of Mechanical Engineering. Univ. of Washington. 3900 Stevens Way NE, Seattle, WA 98195. E-mail: [email protected]

1 16 INTRODUCTION

17 Washington State Ferries (WSF) operates a ﬂeet of 22 vessels in the Puget

18 Sound region of Washington State, USA, and British Columbia, Canada. It is

19 the largest ferry system in the United States and services the highest volume

20 of automobiles of any ferry service in the world. The Seattle-Bremerton route,

21 servicing 2.33 million passengers and 642,000 vehicles per year (Smith 2013), is

22 the primary subject of the current study. In a section of this route named Rich

23 Passage, denoted by the box in Fig. ??, the channel narrows to 600m (?). Due

24 to the narrow channel and steep bathymetry of the region, vessel wake-induced

25 erosion is a concern for the surrounding shoreline.

26 A vessel operating protocol (power limit) of 8.2 m/s (16 knots) is presently

27 in place to reduce wake-induced morphological eﬀects within the channel (Klann

28 2002). The WSF vessels that serve the Seattle-Bremerton route are known as

29 Issaquah Class and Super Class vessels. While of qualitatively similar size, the

30 Super Class vessels are longer, narrower, and heavier than the Issaquah Class

31 vessels. The protocol is applied to the Issaquah Class only, because the Issaquah

32 Class wakes have been classiﬁed, visually, as much larger than the Super Class

33 wakes. This paper seeks to quantify the diﬀerences between the wakes of the two

34 vessel classes through the use of ﬁeld data and an analytical model.

35 Rich Passage has been the subject of a comprehensive wake and shoreline

36 study conducted upon the introduction of the Rich Passage I, a passenger-only

37 fast ferry that served the Seattle-Bremerton route for Kitsap Transit. The study,

38 conducted by Golder and Associates, concluded that the Rich Passage I meets

39 wake height guidelines speciﬁed by

  H < 0.2 T ≤ 3.5 40 H = (1)  −1.4 H < 1.16T T > 3.5

2 41 where H is the the wake height in meters measured 300m from the sailing line of

42 the vessel, and T is the wake period in seconds. This criterion is based on the

43 observation that low-frequency waves contain more erosive potential because of

44 their higher energy ﬂux, and therefore have a more restrictive associated height

45 limit. The morphological shoreline response was largely inconclusive (Cote et al.

46 2013).

47 WSF commissioned a similar study of the car ferry ﬂeet in 1985 to measure

48 vessel wakes (Nece et al. 1985). This study involved Issaquah, Superclass, and

49 Evergreen class models in open water in Puget Sound. No modeling eﬀorts ac-

50 companied this study, and no curve ﬁtting or statistical analysis were performed.

51 The results establish qualitatively that the Super class vessel wakes grow at a

52 much slower rate as vessel speed increases when compared to the Issaquah class.

53 However, noise in wake data and diﬃculties in establishing the buoy-boat dis-

54 tance make the conclusions diﬃcult to quantify. Despite these shortcomings, it

55 was found that the data from this study could be used to supplement the ﬁeld

56 data from the current study. Uncertainty in wake height due to natural varia-

57 tion in time-dependent surface wave conditions dominated uncertainty due to the

58 buoy-boat distances.

59 The more general objective of predicting boat wakes has generally been ap-

60 proached as either a computational ﬂuids problem, or as an empirical quantiﬁ-

61 cation of wave height. Previous modeling eﬀorts of phase resolved vessel wakes

62 have been conﬁned to domains of only a few boat lengths due to computational

63 requirements (Raven 1998), which provide limited information about potential

64 shoreline impact. Sophisticated near-vessel models are primarily used to model

65 wave breaking and wave making resistance for naval architectural applications

66 (Landrini et al. 1999; Huang and Yang 2016). Phase-averaged models have been

67 successfully implemented (Cote et al. 2013), but the resultant wake structure

3 68 is lost. Other phase-resolved models have focused on vessels with transcritical

69 Froude numbers in shallow water (Kofoed-Hansen et al. 1999). Empirical models

70 have been used to great success within the bounds of the space sampled (Sorensen

71 1997; Kriebel et al. 2003; Ng and Bires )

72 The analytic study of ship wakes was ﬁrst considered mathematically in 1887,

73 when Lord Kelvin showed that the disturbance caused by a point pressure source

74 moving across a ﬂuid surface forms a predictable structure which now bears his

75 name (Thomson 1887). Subsequent work focused on the geometry and structure of

76 wakes, with extensions for arbitrary motion of the pressure point (Stoker 1957).

77 Signiﬁcant gains have been made more recently, with results detailing surface

78 velocity ﬁelds (Yun-gang and Ming-de 2006), the inclusion of viscous forces (Lu

79 and Chwang 2007), and ﬁnite vessel dimensions (Darmon et al. 2014; Benzaquen

80 et al. 2014).

81 The impact of boat wakes on shoreline, and mitigation thereof, has been a

82 continuous concern for caretakers of waterways, as well as a subject of many

83 previous studies (Glamore 2008; Aage et al. 2003; Verhey and Bogaerts 1989). In

84 particular, work has been done on the sediment transport associated with vessel-

85 generated waves (Velegrakis et al. 2007).

86 The eﬀects of bow wakes, which are known to have complex hull-dependent

87 forms (Noblesse et al. 2013) are not considered in this study, though results suggest

88 that they may dominate the vessel-generate wave ﬁeld for low Froude number.

89 METHODS

90 Analytical Model

91 The ﬁrst analytic study of wake patterns was published in 1887, when Lord

92 Kelvin showed that the disturbance caused by a point pressure source moving

93 across a ﬂuid surface forms a predictable structure which now bears his name

94 (Thomson 1887). Subsequent work focused on the geometry and structure of

4 95 wakes, with extensions for arbitrary motion of the pressure point (Stoker 1957).

96 A closed form expression for the sea surface height caused by a ﬁnite pressure

97 distribution moving across a ﬂuid surface was recently developed (Benzaquen et al.

98 2014; Darmon et al. 2014). The equations developed by Darmon, Benzaquen, et

99 al. are the basis for the analytical modeling presented below.

100 The analytical model must be conﬁned to potential ﬂow theory in order to

101 yield tenable solutions for the sea surface height. Therefore, the restrictions

102 µ = 0 and ~ω = ∇ × ~u = 0, (2)

103 are required. Here, µ is the viscosity of the flow, ~u is the flow field in Euclidean

104 coordinate space, and ~ω is the vorticity.

105 The sea surface height generated by a moving pressure distribution p(x, y)

106 (Rapha¨eland deGennes 1996) is given by

Z ∞ Z ∞ −i(kxx+kyy) dkxdky kpˆ(kx, ky)e 107 ζ(x, y) = − lim 2 2 2 2 (3) →0 0 0 4π ρ ω(k) − U kx + 2iUkx

108 wherep ˆ(kx, ky) is the Fourier transform of the pressure distribution p(x, y), ρ is the

p 2 2 109 constant ﬂuid density, ω(k) is the surface wave dispersion relation, k = kx + ky,

110 and U is the pressure distribution’s constant forward velocity. The parameter is a

−1 111 variable with units s that approaches zero, ensuring that the radiation boundary

112 condition is satisﬁed. For a derivation of Eq. (3), the reader is referred to Rapha¨el

113 and deGennes (1996). Deﬁne a coordinate system such that the vessel is located

114 at the origin, and is moving in the −x-direction, with the sea surface displacement

115 in the z-direction. The y-axis deﬁnes the neutral sea surface perpendicular to the

116 sailing line of the vessel. Following the method developed by Darmon et al., the

5 117 nondimensionalization conditions for a boat of length b are:

x y X = Y = K = k b b b X x s (4) 118 4π2ζ pˆ b K = k b Z = Pˆ = ˜ = , Y y b ρgb3 g

119 Nondimensionalization yields

Z(X,Y )b Z π/2 Z ∞ dK dK K ρgb3Pˆ(K ,K )e−i(KX X+KY Y ) = − lim X Y b x y 2 2 2 gK K2 g K 4π ˜→0 −π/2 0 4π b ρ 2 X p X b − U b2 + 2i˜ b U b Z π/2 Z ∞ 3 ˆ −i(KX X+KY Y ) dKX dKY Kρgb P (Kx,Ky)e 120 Z(X,Y ) = − lim (5) 4 gK K2 g ˜→0 −π/2 0 b ρ 2 X p b − U b2 + 2i˜ b3 UKX Z π/2 Z ∞ dK dK KPˆ(K ,K )e−i(KX X+KY Y ) = − lim X Y x y ˜→0 2 2 −π/2 0 K − F r KX + 2iF˜ rKX

121 An elliptic transform

˘ ˘ ˘ ˘ −1 ˘ ˘ 122 X = R cosϕ ˘ Y = W R sinϕ ˘ KX = K cos θ KY = W K sin θ, (6)

123 permits solutions for an elliptic Gaussian pressure distribution. Deﬁne an aspect

124 ratio W > 0 to be the ratio of the y to x characteristic dimensions of P (x, y).

125 Applying this transformation,

Z(R,˘ ϕ˘) = p Z π/2 Z ∞ ˘ ˘ 2 ˘ −2 2 ˘ ˆ˘ ˘ −iK˘ R˘(cosϕ ˘ cos θ˘+sinϕ ˘ sin θ˘) K ˘ ˘K cos θ + W sin θP (K)W e 126 − lim dKdθ p , ˜→0 −π/2 0 W cos2 θ˘ + W −2 sin2 θ˘ − F r2K˘ 2 cos2 θ˘ + 2iF˜ rK˘ cos θ˘ (7)

K˘ ˆ˘ ˘ 127 where the Jacobian of the coordinate transformation in Eq. (6) is W , and P (K) =

Pˆ(KX ,KY ) 128 W to account for the reduced dimension. Simplifying,

6 Z(R,˘ ϕ˘) =

129 p ˘ ˘ ˘ ˘ (8) Z π/2 Z ∞ W 2 cos2 θ˘ + sin2 θ˘Pˆ(K˘ )e−iKR cos(θ−ϕ˘)Kd˘ Kd˘ θ˘ − lim p . ˜→0 −π/2 0 W 2 cos2 θ˘ + sin2 θ˘ − W F r2K˘ cos2 θ˘ + 2iF˜ rW cos θ˘

130 The Sokhotski-Plemelj formula (Appel 2007) states that for a function f(x) with

131 Cauchy principal value Q,

Z b f(x) Z b f(x) 132 lim dx = ∓iπf(0) + Q dx. (9) ˜→0 a x ± i˜ a x

133 Eq. (8) is manipulated into this form using

p W 2 cos2 θ˘ + sin2 θ˘ − W F r2K˘ cos2 θ˘ F r cos θ˘ x = dx = − dK˘ 2F rW cos θ˘ 2 134 (10) p ˘ ˘ ˘ ˘ W 2 cos2 θ˘ + sin2 θ˘Pˆ(K˘ )e−iKR cos(θ−ϕ˘) f(x) = K˘ . F r2W cos2 θ˘

135 The zero point in f is determined uniquely by K˘ since it is the only independent

136 variable. Therefore, designate

p 2 2 ˘ 2 ˘ ˘ W cos θ + sin θ 137 K0 ≡ (11) W F r2 cos2 θ˘

138 by solving with x = 0 in K˘ . It is argued in Benzaquen et al. (2014) that R b 139 Q a f(x)dx/x is a rapidly decreasing function that can be ignored. Therefore,

140 the sea surface height is given by

˘ ˆ˘ ˘ 2 2 ˘ 2 ˘ −iK0R cos(θ−ϕ˘) Z π/2 P (K0) W cos θ + sin θ e 141 Z(R,˘ ϕ˘) ≈ iπ dθ.˘ (12) 2 4 4 −π/2 W Fr cos θ˘

142 The pressure distribution is assumed to be of the form

7 2 2 πmg −( xπ ) −( yπ ) 143 p(x, y) = e b W b . (13) W b2

144 The leading coeﬃcient is chosen such that the integral of the pressure function

145 Eq. (13) over the plane of the sea surface is equal to the total force of the vessel

146 on the water, mg (i.e., p(x, y) has a net ’weight’ equal to that of the vessel.)

147 The b and W parameters can be tuned to approximate the geometry of the

148 physical hulls. To that end, p(x, y) may be converted into an equivalent ζ(x, y)

149 to represent the sea surface elevation at which that pressure is experienced:

p(x, y) 150 ζ(x, y) = . (14) ρg

151 The volume between ζ(x, y) and the xy axis deﬁnes a 3-dimensional ﬁgure, S that

152 can be geometrically compared to a CAD hull model, C. In subsequent analysis, R 153 the W and b parameters are chosen such that the absolute norm |S − C| dV is R3

154 minimized. The pressure distribution then matches the weight of the vessel and

155 the shape is tuned to match that of the physical hull, within the constraints of

156 the analytical model.

157 The physical pressure distribution is then converted into a form compatible

158 with Eq. (12). In the nondimensional form speciﬁed by Eq. (4),

2 πm −(Xπ)2−( Y π ) 159 P (X,Y ) = e W . (15) W ρb3

160 After a Fourier transform,

m 1 2 2 2 − 2 (K +W K ) 161 Pˆ(K ,K ) = e 4π X Y . (16) X Y 2πρb3

8 162 And ﬁnally, applying the elliptic transform Eq. (6),

m K˘ 2 ˘ − 2 163 Pˆ(K˘ ) = e 4π . (17) 2πρb3

164 Measurements

165 On 17 September 2014 and 18 November 2014, the wakes of the ferry boats

166 servicing the Seattle-Bremerton route were measured via data buoys deployed

167 from R/V Jack Robertson. An approximate sailing line for an approaching ferry

168 was obtained from Automatic Identiﬁcation System (AIS) vessel tracking data.

169 Buoys were then deployed in a line running approximately perpendicular to the

170 predicted sailing line of the ferry. The ferry crossing time was recorded as the

171 time when the AIS coordinates of the ferry intersected the time-dependent best

172 ﬁt line of the array of buoys.

173 The data buoys in use consisted of four SWIFT and eight µSWIFT buoys

174 (Thomson 2012). These buoys are free-ﬂoating spar-type buoys equipped with

175 GPS loggers. The µSWIFT uses a QStarz BT-Q1000eX and the SWIFT uses a

176 Microstrain 3dm-35. Data were sampled at rates varying from 4 Hz to 10 Hz.

177 The GPS logs both positional velocimitry data, but the horizontal positional ac-

178 curacy of commercial GPS devices such as this one are only on the order of 10m–

179 signiﬁcantly larger than is required to phase-resolve individual waves. The rel-

180 ative horizontal velocity resolution, however, is approximately 0.05 m/s through

181 Doppler phase processing of the raw signals (Herbers et al. 2012; Thomson 2012).

182 The vertical elevation and velocity do not have suﬃcient accuracy to be consid-

183 ered for data processing. Therefore, the positional GPS data are used to obtain

184 approximate locations and trajectories of the buoys, and the horizontal velocity

185 data are used to capture the motion of the sea surface. Typically, buoy motion

186 due to ocean current was less than 1 m/s.

9 187 In linear wave theory, the motion of inertial particles inﬂuenced by sinusoidal

188 wave motion is given by

kz 189 v(t) = aωe cos θ(t) (18)

190 for wave amplitude a, angular frequency ω, wavenumber k, and distance below

191 the surface z. Similarly, the surface elevation is given by

192 ζ(t) = a cos θ(t). (19)

193 For a particle on the surface, the sea surface height may be obtained from the

194 velocity via the relation v(t) 195 ζ(t) = . (20) ω(t)

196 The natural frequency of both types of drifter buoy is well above the frequencies

197 of the observed wake pattern. As such, the drifters are treated as inertial surface

198 particles. The goal of the time domain analysis is, therefore, to obtain accurate

199 and well-behaved values for ω and v(t) in order to reconstruct the sea surface

200 height. Here, well-behaved refers to avoiding low-frequency velocity drift and

201 artiﬁcially small values of ω.

202 The surface reconstruction is performed by a process of

203 • Calculating and subtracting low-pass ﬁltered velocity vectors, in order to

204 remove the eﬀects of buoy drift

205 • Obtaining a scalar velocity by calculating the principal axes for buoy kinetic

206 energy in the plane of the ocean and taking only the velocity of the major

207 axis

208 • Calculating a local wave frequency ω by calculating the zeros in the scalar

209 velocity

210 • Applying Eq. (20) to obtain the sea surface height. The conversion from

10 211 scalar velocity to sea surface height is shown in Fig. ??

212 In a rough sense, the surface disturbance caused by a passing ferry boat can

213 be expressed as a near-Dirac disturbance, or equivalently, a broadband wave gen-

214 eration. Denote such a function

Z ∞ ikx 215 ζ(x, 0) = a(k)e dk (21) −∞

216 where a(k) speciﬁes a magnitude associated with a given wavenumber k. Then, √ 217 for the outward spreading of this disturbance, the dispersion relation ω(k) = gk

218 is used. This is the deep water limit, which for the shallowest in-situ depth

219 encountered of 20 meters, is permissible for wavelengths of less than 40 meters.

220 The time-dependent sea surface height is then

Z ∞ √ i(kx− gkt) 221 ζ(x, t) = a(k)e dk. (22) −∞

222 In the limit of large x and t (as in the case of the ﬁeld data), a stationary

223 phase approximation (Stoker 1957) may be invoked in order to evaluate Eq. (22),

224 yielding r gt2 π 225 ζ(x, t) ≈ a(k ) cos k (x) − ω(k )t ± (23) 0 πx3 0 0 4 √ 2 2 226 where k0 = gt /(4x ) solves the equation ∂/∂k kx − gkt = 0.

227 Therefore, for a given (x, t) displacement from the initial disturbance (i.e., a

228 time t after a vessel’s closest approach to a point x of observation), the observed

229 frequency of waves should be given by

gt 230 ω(x, t) = . (24) 2x

231 Note that in Eq. (24) the expression for velocity is the group velocity, g/(2ω),

11 232 rather than the phase velocity. For constant x = d (i.e. a buoy that is stationary

233 in the reference frame of the ocean surface) the observed frequency should in-

234 crease linearly in time with slope g/(2d). Comparisons of predicted and observed

235 wave frequencies (Fig. ?? being a typical case) show that treating the vessels as

236 generating broadband wave pulses serves as a good predictor for the arrival of

237 wave frequencies observed by the buoys. This is a useful check in cases where the

238 distance from sailing line is diﬃcult to determine.

239 RESULTS

240 Analytical Model

241 The analytical model described by Eq. (12) and Eq. (13) produces a steady

242 state sea surface height as a function of boat length b, boat aspect ratio W , boat

243 mass m, water density ρ, and the boat velocity. If Eq. (12) is expressed with the

244 leading coeﬃcients in dimensional form, the expression becomes

˘  K0 ˘  − 2 2 2 ˘ 2 ˘ −iK0R cos(θ−ϕ˘) m Z π/2 e 4π W cos θ + sin θ e 245 ζ(x, y) = Im  dθ˘ (25) 2 2  2 4 4  4π ρb −π/2 W Fr cos θ˘

246 One may conclude, then, that the sea surface height scales linearly as the boat

247 mass, and inversely as the water density. Eq. (25) contains a complex dependence

248 on the remaining variables. These remaining dependencies are shown graphically

249 in Fig. ??. The sea surface forms a distinctive Kelvin wake pattern. In Fig. ??,

250 two such solutions are shown, for diﬀerent values of W . At low W , the largest

251 waves are located at the intersection between the angled diverging wake and the

252 approximately horizontal transverse wake. At W = 1, indicating a circular pres-

253 sure distribution, the greatest perturbation in the sea surface is located directly

254 behind the vessel, in the centers of the transverse wake waves. The growth of the

255 diverging wake relative to the transverse wake is typically associated with a larger

12 256 vessel Froude number (higher velocity, or smaller vessel length when all else is

257 held constant). The eﬀects of this transition are further explored on page 16.

258 The plots in Fig. ?? explore the maximum wake height, as this is the relevant

259 parameter under study for the ferry boats. Maximum wake height is deﬁned to

260 be the absolute value of the greatest deformation in the free surface, at a given

261 perpendicular distance d from the axis of vessel motion. Of the variables other

262 than m and ρ, the maximum wake height is determined by the Froude number and

263 the aspect ratio W . However, because the Froude number contains two physically

264 relevant variables, velocity and characteristic length, their independent eﬀects

265 are explored. In each case, in Fig. ??, the vessel mass and water density are

266 unchanged.

267 In order to address the physical wakes, the model parameters were chosen to

268 represent an Issaquah class and a Super class vessel. Using the available vessel

269 dimensions, mass, and CAD models, Gaussian pressure distributions of the form in

270 Eq. (13) were created. The resulting wake proﬁles, shown in Fig. ?? and generated

271 from the values in Table ??, were highly dependent on the boat length b, but in all

272 cases, the Issaquah class model produced much larger wakes than the Super class

273 model. The presented models match velocity rather than Froude number, for dual

274 reasons. The ferry operating conditions are determined by the time schedule, and

275 so the same velocity must be attained when comparing performance. In addition,

276 the Froude number of the Issaquah class is larger by a small enough amount that

277 slowing the vessel to attain Froude number parity would still not eliminate the

278 diﬀerence in wake height. The wake proﬁles, presented at the same scale, indicate

279 signiﬁcantly smaller wake for the Super Class, as well as a weaker diverging wake

280 relative to the transverse wake.

13 281 Measurements

282 A total of 19 ferry crossings were recorded, yielding 79 useful wake histories

283 from the drifter buoy data. These wake histories represent 65 measurements of

284 the Issaquah class at a variety of vessel speeds, and 14 measurements of the Super

285 class, all at approximately its standard operating speed of 18 knots.

286 For one wake event, a µSWIFT and a SWIFT buoy were placed within 10m of

287 each other in order to compare buoy performance. In general, the SWIFT buoy

288 provides less noisy data, but the comparison in Fig. ?? suggests that both varieties

289 of buoy have similar spectra at the frequencies of interest.

290 The raw buoy velocities also indicate close agreement. However, the sea surface

291 height reconstruction is highly sensitive to the exact recorded velocities. Despite

292 qualitatively similar buoy velocities, the ﬁnal sea surface height diﬀers consider-

293 ably between the two measurements. This reﬂects the highly localized nature of

294 sea surface velocities. The spread in maximum sea surface height at consistent

295 experimental conditions (see Fig. ??) is qualitatively similar for both buoy types

296 when the aggregate data are taken. For this reason, data from both buoy types

297 are used interchangeably.

298 The angular frequency associated with the maximum wake amplitude for a

299 wake event was recorded. In general, the observed frequency was lower at larger

300 distances from the sailing line, as the dispersion relation for deep water waves

301 would suggest. However, the relationship between angular frequency and max-

302 imum wake height, Fig. ??, demonstrated a trend that appears to depend on

303 the vessel speed. The most energetic waves are the ones with low angular fre-

304 quency and large maximum height. Therefore, it is particularly concerning that

305 the largest wakes associated with higher vessel speeds are also the lowest frequency

306 wakes.

14 307 Comparisons

308 Because one of the primary concerns for WSDOT ferry wakes is the overtop-

309 ping of sea walls, and the shoreline morphology is quite steep, the maximum sea

310 surface height is considered as a metric for comparison. Maximum sea surface

311 height versus distance from sailing line was compared across all measurements

312 and models. The analytical model’s parameters were tuned to match the shape

313 and mass of the Issaquah and Super Class hulls, as described on page 4. Fig. ??

314 displays the overall comparison between the various wake measurements studied.

315 The analytical model does a good job of capturing the decay in wake height with

316 increased distance from the sailing line that is observed in the ﬁeld data. At low

317 vessel speed, however, the analytical model tends to predict a smaller disturbance

318 than is physically observed.

319 Another method of visualizing the model’s predictive ability is to treat each

320 buoy as an inertial particle moving on the surface of the ocean, in the reference

321 frame of the vessel. The buoy’s location is converted to a time varying displace-

322 ment from the vessel, and the analytic solution is calculated along the transect

323 the buoy takes, parameterized by time. The parameterized analytical solution

324 ζ(t) then mimics the history of the sea surface height experienced by a numerical

325 buoy taking the same trajectory as the buoy in the ﬁeld. This permits direct

326 comparisons with reconstructed sea surfaces from the wave buoys.

327 Several such comparisons are presented in Fig. ??. This method is quite sensi-

328 tive to clock synchronization between the vessel and buoy, as well as vessel speed.

329 For each comparison, the vessel class and speed are matched between the ana-

330 lytical model and the observations for the measured vessel crossing. The initial

331 wake structure is well-captured, though the timing may have an oﬀset due to

332 ocean currents and the exact location of the AIS receiver on the ferry. The wave

333 packeting that is often seen by the physical buoys was not reliably reproduced by

15 334 the model, and the reasons for this are explored on 16.

335 DISCUSSION

336 Models and data indicate that at similar operating speeds, the Issaquah Class

337 vessels make signiﬁcantly larger wakes than the Super Class vessels. This result is

338 somewhat unintuitive, as the Super Class is a longer and heavier vessel. Results

339 from the analytical model point to a dual justiﬁcation for the wake height disparity.

340 The increased vessel length for the Super Class has the eﬀect of reducing the vessel

341 Froude number, and the decreased aspect ratio W also works to decrease the wake.

342 These two eﬀects counteract the increased weight to yield an overall lower wake.

343 Furthermore, wake height reduction is achieved by reducing the vessel speed

344 for the Issaquah Class. The Super Class data do not include signiﬁcant speed

345 variation, but the data displayed in Fig. ?? demonstrate that the full-speed Su-

346 perclass produces much smaller peak wake height than the full-speed Issaquah,

347 and qualitatively comparable wake heights to the speed-limited Issaquah class.

348 The ﬁeld data support the conclusions obtained in the 1985 study (Nece et al.

349 1985). Analytic data indicate a roughly linear relationship between peak wake

350 height at a given distance from the sailing line and vessel speed in the regime of

351 interest. Therefore, in assessing a given distance and peak height objective, the

352 ﬁeld data plots should be used as two data points, with a linear interpolation to

353 determine appropriate speed.

354 At low Froude numbers, the transverse wake dominates the wake ﬁeld of a ves-

355 sel. However, ﬁeld observations and CFD modeling have shown that, particularly

356 at low Froude numbers, the vessels produce a signiﬁcant bow wake. The bow wake

357 is a consequence of the physical displacement of the water in front of the vessel

358 and is not captured by the model. Particularly at the lowest operational speeds

359 of the ferries, the bow wake dominates the waves produced by the vessel. This is

360 the proposed mechanism for the model failure at low Froude number (below 0.35).

16 361 In addition, the packeting observed in Fig. ?? may be a beat frequency produced

362 by the interaction of the bow wake and the Kelvin wake. At F r > 0.35, the bow

363 wake was not observed to aﬀect the maximum wake height measurements.

364 CONCLUSIONS

365 The wakes produced by Issaquah and Super Class WSF vessels were studied

366 using an array of GPS buoys and with an analytic potential ﬂow wake model.

367 Each of these methods produced results that indicate a maximum wake height

368 for a given distance from the sailing line of the vessel and a given vessel speed.

369 These results are summarized in Fig. ??. The in-situ buoy measurements are

370 regarded to be the most reliable, but also the contain the most sparse data, as

371 well as signiﬁcant noise. A higher-ﬁdelity sea surface height reconstruction may

372 be possible using vertical acceleration, given a high enough sampling rate. The

373 4-10 Hz sampling rate used presently was insuﬃcient to accurately represent buoy

374 heave.

375 Despite being an older and larger vessel, the Super Class ferry produces smaller

376 wakes than the Issaquah class at a given speed. The relatively simple analytic

377 representation of these vessels is able to attribute diﬀerence in wake height to the

378 geometric qualities of aspect ratio and vessel length, as well as vessel mass. The

379 measurements and subsequent model comparison constitute a novel approach the

380 question of vessel wake management.

17 381 NOTATION

Variable Name Description ρ Seawater density. Assigned a value of 1025kg/m3 for Puget Sound. ω Angular frequency of a wave. {x, y, z} Cartesian coordinates in the frame of the ferry boat. x denotes the streamwise direction, y is the crossstream direction, and z is the vertical direction. The origin of the grid is located at the mass centroid in x and y, and at the flat sea surface in z. kx, ky Wavenumbers associated with the analytic wake model. W Aspect ratio of a ferry boat in the analytic wake model. This is the ratio of the major to minor axis of the ellipse representing the boat. A smaller W implies a more streamlined vessel. b Characteristic length of the ferry. √ F r Length Froude number, defined by U/( gb). U Free stream velocity of flow past the ferry boat. ζ Value of the sea surface elevation, measured as a signed perturbation from a flat surface of value 0. Z Nondimensional sea surface, defined by 4π2ζ/b. g Gravitational acceleration, given a value of 9.8m/s2. h Maximum wake height. This is the maximum of the absolute value of the perturbation in the free surface. a Wave amplitude, for the purpose of linear wave analysis.

18 382 REFERENCES

383 Aage, C., Bell, A., Bergdahl, L., Blume, A., Bolt, E., Eusterbarkey, H., Hiraishi,

384 T., Kofoed-Hansen, H., Maly, D., Single, M., Rytkonen, J., and Whittaker, T.

385 (2003). Guidelines for Managing Wake Wash From High-Speed Vessels.

386 Appel, W. (2007). Mathematics for Phyiscs and Physicists. Princeton University

387 Press.

388 Benzaquen, M., Darmon, A., and Raphael, E. (2014). “Wake pattern and wave

389 resistance for anisotropic moving disturbances.” Physics of Fluids.

390 Cote, J., Curtiss, G., and Osbourne, P. (2013). “Rich passage I wake validation

391 and shore response study.” Ports.

392 Darmon, A., Benzaquen, M., and Raphael, E. (2014). “Kelvin wake pattern at

393 large froude numbers.” J. Fluid Mechl.

394 Glamore, W. C. (2008). “A Decision Support Tool for Assessing the Impact of

395 Boat Wake Waves on Inland Waterways.” International Conference on Coastal

396 and Port Engineering in Developing Countries, (1), 20.

397 Herbers, T., Jessen, P., Janssen, T., Colbert, D., and MacMahan, J. (2012).

398 “Observing ocean surface waves with gps-tracked buoys.” J. Atmos. Oceanic

399 Technol., 29, 944–959.

400 Huang, F. and Yang, C. (2016). “Hull form optimization of a cargo ship for reduced

401 drag.” Journal of Hydrodynamics, Ser. B, 28(2), 173–183.

402 Klann, A. (2002). “Property owners, state, settle fast ferry lawsuit.” Ha-

403 gens Berman Sobol Shapiro LLP,

404 case/pressrelease/closed-case-property-owners-state-settle-fast-ferry-lawsuit>.

405 Kofoed-Hansen, H., Jensen, T., Kirkegaard, J., and Fuchs, J. (1999). “Prediction

406 of wake wash from high-speed craft in coastal areas.” Proc. Conf. Hydrodynam-

407 ics of High Speed Craft, 24–25.

408 Kriebel, D., Seelig, W., and Judge, C. (2003). “Development of a uniﬁed descrip-

19 409 tion of ship-generated waves (presentation).” 1–37.

410 Landrini, M., Colagrossi, A., and Tulin, M. (1999). “Breaking bow and stern

411 waves: Numerical simulations.” 15th Int. Workshop on Water Waves and Float-

412 ing Bodies.

413 Lu, D. and Chwang, A. (2007). “Interfacial viscous ship waves near the cusp lines.”

414 Wave Motion, 44(7-8), 563–572.

415 Nece, R., McCaslin, M., and Christensen, D. (1985). “Ferry wake study.” Report

416 No. WA-RD-70.1, Washington State Dept. of Transportation.

417 Ng, M. and Bires, R. “Vessel wake study.

418 Noblesse, F., Delhommeau, G., Liu, H., Wan, D., and Yang, C. (2013). “Ship bow

419 waves.” Journal of Hydrodynamics, 25(4), 491–501.

420 Rapha¨el,E. and deGennes, P. (1996). “Capillary gravity waves caused by a moving

421 disturbance.” Phys. Rev. E.

422 Raven, H. (1998). “Inviscid calculations of ship wave making — capabilities, lim-

423 itations and prospects.” Proc. 22nd Symposium on Naval Hydrodynamics.

424 Smith, B. (2013). “It’s a boat, its a train it’s super-transit!.” Multimodal Opera-

425 tions Planning Workshop.

426 Sorensen, R. (1997). “Prediction of Vessel-Generated Waves with Reference to

427 Vessels Common to the Upper Mississippi River System.” (December 1997),

428 50.

429 Stoker, J. (1957). Water Waves; the Mathematical Theory with Applications. In-

430 terscience, New York.

431 Thomson, J. (2012). “Wave breaking dissipation observed with swift drifters.” J.

432 Atmos. Oceanic Technol., 29.

433 Thomson, W. (1887). “On the waves produced by a single impulse in water of any

434 depth.” Proc. R. Soc. London Ser. A42, 80–83.

435 Velegrakis, A. F., Vousdoukas, M. I., Vagenas, A. M., Karambas, T., Dimou, K.,

20 436 and Zarkadas, T. (2007). “Field observations of waves generated by passing

437 ships: A note.” Coastal Engineering, 54(4), 369–375.

438 Verhey, H. and Bogaerts, M. (1989). “Ship waves and the stability of armour lay-

439 ers protecting slopes.” 9th International Harbour Congress, Antwerp, Belgium,

440 Delft Hydraulics.

441 Yun-gang, W. U. and Ming-de, T. A. O. (2006). “An Asymptotic Solution of

442 Velocity Field in Ship Waves.” Journal of Hydrodynamics, 18(4), 387–392.